Euler Equation

We consider the following optimization problem

subject to

where we assume that and .

We convert it into the following Bellman equation

where

Now we calculate Euler equation.

(1) FOC w.r.t. k’:

or

(2) Envelope condition(FOC w.r.t. k):

(3) Shift the envelope condition ahead by one period:

(4) Substitute the first one into this:

Thus

So far, we have almost the same form as yesterday’s.

When , it is going to be

At steady state, and and .

So

That is

Thus

On the other hand, from FOC w.r.t. ,

since .

Recall that the derivative of logarithm is reciprocal of inside the logarithm times

the derivative of the term inside.

Thus

where we have to solve it for together with above simultaneously.